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Difference between revisions of "2022 AMC 10B Problems"
(→‎Problem 8)
Line 59: Line 59:
 
Consider the following <math>100</math> sets of <math>10</math> elements each:
 
Consider the following <math>100</math> sets of <math>10</math> elements each:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
&\{1,2,3,\cdots,10\}, \\
+
&\{1,2,3,\ldots,10\}, \\
−
&\{11,12,13,\cdots,20\},\\
+
&\{11,12,13,\ldots,20\},\\
−
&\{21,22,23,\cdots,30\},\\
+
&\{21,22,23,\ldots,30\},\\
 
&\vdots\\
 
&\vdots\\
−
&\{991,992,993,\cdots,1000\}.
+
&\{991,992,993,\ldots,1000\}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  

Revision as of 15:41, 17 November 2022

2022 AMC 10B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

Define $x\diamondsuit y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamondsuit(2\diamondsuit3))-((1\diamondsuit2)\diamondsuit3)?\] $\textbf{(A)}\ {-}2 \qquad\textbf{(B)}\ {-}1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$

Solution

Problem 2

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 3

How many three-digit positive integers have an odd number of even digits?

$\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

Solution

Problem 4

A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $700$th hiccup occur?

$\textbf{(A) }15 \text{ seconds after } 4:58$

$\textbf{(B) }20 \text{ seconds after } 4:58$

$\textbf{(C) }25 \text{ seconds after } 4:58$

$\textbf{(D) }30 \text{ seconds after } 4:58$

$\textbf{(E) }35 \text{ seconds after } 4:58$

Solution

Problem 5

What is the value of \[\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?\] $\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt{15} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{105}$

Solution

Problem 6

How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?

$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

Solution

Problem 7

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16$

Solution

Problem 8

Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*}

How many of these sets contain exactly two multiples of $7$?

$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 43\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 50$

Solution

Problem 9

The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!}\]can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$?

$\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

Solution

Problem 10

Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?

$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13$

Solution

Problem 11

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 12

A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$?

$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$

Solution

Problem 13

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 14

Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \cdots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain?

Solution

Problem 15

Let $S_n$ be the sum of the first $n$ term of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$?

$\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$

Solution

Problem 16

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 17

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 18

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 19

XXX

$\textbf{(A)}\ 14 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 22 \qquad\textbf{(D)}\ 26 \qquad\textbf{(E)}\ 30$

Solution

Problem 20

Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$?

$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$

Solution

Problem 21

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 22

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?

$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

Solution

Problem 23

Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?

$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$

Solution

Problem 24

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

Problem 25

XXX

$\textbf{(A)}\ X \qquad\textbf{(B)}\ X \qquad\textbf{(C)}\ X \qquad\textbf{(D)}\ X \qquad\textbf{(E)}\ X$

Solution

See also

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by
2021 Fall AMC 10B Problems 		Followed by
2023 AMC 10B Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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